D-the Distance From The Geometric Parameters Are Graphs And Authentication Code

In this paper,we study the d-bounded distance-regular graphΓwith diameter d．We obtain two new formula of calculation of subspace and construct a class of Cartesianauthentication code using the subspaces．The main results are as follows．1．LetΓbe a d-bounded distance-regular graph with geometric parameters(d,b,α)and d≥3．Supposeâ–³(?)â–³'and i+1≤i+s≤i+s+t,and supposeâ–³andâ–³' aresubspaces with diameter i and i+s+t,respectively．Then the numbers of subspaces (?) inP(x) withâ–³(?)(?)(?)â–³' and diameter i+t,denoted by N(i,i+sï¼›i+s+t),is determinedby i,s and t,independent of the choice ofâ–³andâ–³' and is given bywhere(?)_b² is the Gaussian coefficents with basis b²ï¼Ž2．LetΓbe a d-bounded distance-regular graph with geometric parameters(d,b,α)and d≥3．For x∈V(Γ),let P(x) be a set of all subspaces containing x inΓ．Supposeâ–³₁(?)â–³,0≤t≤i+t and j+t≤i+j+t≤d,and supposeâ–³₁ andâ–³are subspacewith diameter t and i+t,respectively．Then the number of subspacesâ–³' in P(x) withdiameter j+t such that△∩△'=â–³₁,denoted by M₁(t,i+t,j+tï¼›d),is determined byi,j and t,independent of the choice ofâ–³₁ andâ–³and is given byFurthermore,the number of subspacesâ–³' in P(x) with diameter j+t such thatd(△∩△')=t,denoted by M(t,i+t,j+tï¼›d),is determined by i,j and t,independent ofthe choice ofâ–³₁ andâ–³and is given bywhere(?)_b² is the Gaussian coefficents with basis b²ï¼Ž3．LetΓbe a d-bounded distance-regular graph with geometric parameters(d,b,α)and d≥3．For x∈V(Γ),let P(x)be a set of all subspaces containing x inΓ．We also suppose 1≤m₀0 is a fixed subspace with diameter m₀ in P(x)．Definethe source states to be the subspaces with diameter m containingâ–³₀．Define the encodingrules to be the subspaces with diameter d-m₀,which intersectsâ–³₀ at{x}．Define themessages to be the subspaces with diameter m-m₀,which intersectâ–³₀ at {x}．Denotethe set of source states,the set of encoding rules,and the set of messages by S,εandM,respectively.Given any△∈S and anyâ–³₁∈ε,f(â–³,â–³₁)=△∩△₁ is defined．Thenwe confirmed that it is a Cartesian authentication code．4．The construction above yields an Cartesian authentication code,whose size pa-rameters arewhere (?)_b² is the Gaussian coefficents with basis b²ï¼ŽAssume that the encoding rules are chosen according to a uniform probability dis-tribution,and denote the probabilities of a successful impersonation attack and of asuccessful substitution attack by P_I and P_S,respectively．Then...